Julien Gaboriaud, Dépt. de physique et CRM

Title: A commutant approach to the Racah, Bannai-Ito and Askey-Wilson algebras.

Abstract: The Racah, Bannai-Ito (BI) and Askey-Wilson (AW) algebras are respectively known as the centralizers of the su(1,1), osp(1|2), U_q(su(1,1)) algebras in their respective threefold tensor product. In this talk, it will be shown how each of these pictures can be given a dual description in the sense of Howe. More precisely, as a dual picture of the above, the Racah algebra will be obtained as the commutant of a o(2) oplus o(2) oplus o(2) subalgebra of o(6) in the oscillator representation of U(o(6)). This result will be explained by highlighting the Howe dual pair (su(1,1), o(6)) that comes into play. Similar dual pictures will then be presented for the BI algebra and the AW algebra. Whether such dual pictures exist for the higher rank generalizations of the Racah, BI and AW algebras will be discussed at the end, along with some open questions.